GoKawiilOpenAI system produces proof of finite-time singularity in Navier–Stokes equations
OpenAI says an internal AI model, more capable than GPT-6 Astra, generated a solution to part of the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. The proof demonstrates that three-dimensional incompressible fluid dynamics governed by these equations can develop a singularity—unbounded speed growth—in finite time, even from smooth initial conditions. OpenAI released both a written explanation of the proof and a formal verification built in the Lean proof assistant.
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The Navier–Stokes problem has stumped mathematicians for about 90 years and underpins fields like aerodynamics, meteorology, and biomedical flow modeling, so any credible progress carries real scientific weight. That an unreleased AI model reportedly generated this result signals a notable jump in machine reasoning capability for advanced mathematics, and the choice to publicize it before the model's public release suggests OpenAI wants to shape expectations about how fast AI is progressing.
- OpenAI claims an internal AI system solved a portion of the Navier–Stokes Millennium Prize Problem.
- The proof shows fluid motion equations can produce a finite-time singularity despite viscosity's smoothing effect.
- OpenAI published a proof writeup and a Lean-based formal verification, and says the model used exceeds GPT-6 Astra's capabilities.
The Millennium Problems by Keith Devlin — If a claimed proof of Navier–Stokes existence and smoothness has you curious about what's actually at stake, Keith Devlin's book walks through all seven Millennium Prize Problems in plain language. It's a great way to understand why fluid singularities have stumped mathematicians for roughly 90 years, no PhD required.
See The Millennium Problems by Keith Devlin on Amazon → Affiliate link — we may earn a commission on purchases, at no extra cost to you. Product picked by AI based on this article; it is not a tested recommendation.Source: openai.com, 2026-09-08
Published there as: “On the Navier–Stokes Millennium Prize Problem”
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